A universal group-theoretic characterisation of $p$-typical Witt vectors

Abstract

For a prime $p$ and a commutative ring $R$ with unity, let $W(R)$ denote the group of $p$-typical Witt vectors. The group $W(R)$ is endowed with a Verschiebung operator $V:W(R)\to W(R)$ and a Teichmüller map $⟨⟩:R\to W(R)$. One of the properties satisfied by $V,⟨⟩$ is that the map $R\to W(R)$ given by $x\mapsto V⟨x^p⟩-p⟨x⟩$ is an additive map. In this paper we show that for $p\ne 2$, this property essentially characterises the functor $W$. Unlike other characterisations, this is a group-theoretic characterisation, in the sense that it does not use the ring structure of $W(R)$. Most constructions of the group of $p$-typical Witt vectors of non-commutative rings do not have a ring structure, and hence the above characterisation is more suitable for generalisation to the non-commutative setup.

arXiv commentRevised version(8 pages). The non-commutative part of the previous version has been removed due to an error